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''on an  $  n $-
dimensional manifold  $  M $, 
fibration of spin-frames''

A principal fibre bundle  $  \widetilde \pi   :  \widetilde{P}  \rightarrow M $
over  $  M $
with structure group  $   \mathop{\rm Spin} _ {n} $(
see [[Spinor group|Spinor group]]), covering some principal fibre bundle  $  \pi :  P \rightarrow M $
of co-frames with structure group  $   \mathop{\rm SO} _ {n} $. 
The latter condition means that there is given a surjective homomorphism  $  \kappa :  \widetilde{P}  \rightarrow P $
of principal fibre bundles, which is the identity on the base and is compatible with the natural homomorphism  $  \rho :   \mathop{\rm Spin} _ {n} \rightarrow  \mathop{\rm SO} _ {n} $. 
One says that the spinor structure  $  ( \widetilde \pi   , \kappa ) $
is subordinate to the Riemannian metric  $  g $
on  $  M $
defined by  $  \pi $. 
From the point of view of the theory of  $  G $-
structures, a spinor structure is a generalized  $  G $-
structure with structure group  $  G =  \mathop{\rm Spin} _ {n} $
together with a non-faithful representation  $  \rho :   \mathop{\rm Spin} _ {n} \rightarrow  \mathop{\rm SO} _ {n} $(
cf. [[G-structure| $  G $-
structure]]).

In a similar way one defines spinor structures subordinate to pseudo-Riemannian metrics, and spinor structures on complex manifolds subordinate to complex metrics. Necessary and sufficient conditions for the existence of a spinor structure on  $  M $
consist of the orientability of  $  M $
and the vanishing of the Stiefel–Whitney class  $  W _ {2} ( M) $. 
When these conditions hold, the number of non-isomorphic spinor structures on  $  M $
subordinate to a given Riemannian metric coincides with the order of the group  $  H  ^ {1} ( M, \mathbf Z ) $(
see [[#References|[6]]]).

Let  $  C $
be the complexification of the [[Clifford algebra|Clifford algebra]] of  $  \mathbf R  ^ {n} $
with quadratic form  $  q= \sum _ {i=} 1  ^ {n} x _ {i}  ^ {2} $. 
Then  $  C $
has an irreducible representation in a space  $  S $
of dimension  $  2 ^ {[ n/2] } $, 
which defines a representation of  $   \mathop{\rm Spin} _ {n} \subset  C $
in  $  S $. 
Every spinor structure  $  \widetilde \pi   $
on  $  M $
yields an associated vector bundle  $  \pi _ {S} :  S( M) \rightarrow M $
with typical fibre  $  S $, 
called a spinor bundle. The Riemannian connection on  $  M $
determines in a canonical way a connection on  $  \pi _ {S} $. 
On the space  $  \Gamma ( S) $
of smooth sections of  $  \pi _ {S} $(
spinor fields) there acts a linear differential operator  $  D $
of order  $  1 $, 
the Dirac operator, which is locally defined by the formula

$$ 
Du  =  \sum _ { i= } 1 ^ { n }  s _ {i} \cdot \nabla _ {s _ {i}  }
u ,\  u \in \Gamma ( S) ,
$$

where  $  \nabla _ {s _ {i}  } $(
$  i= 1 \dots n $) 
are the covariant derivatives in the directions of the system of orthonormal vector fields  $  s _ {i} $
and the dot denotes multiplication of spinor fields by vector fields which correspond to the above  $  C $-
module structure on  $  S $.

Spinor fields in the kernel of  $  D $
are sometimes called harmonic spinor fields. If  $  M $
is compact, then  $   \mathop{\rm dim}   \mathop{\rm ker}  D < \infty $, 
and this dimension does not change under conformal deformation of the metric [[#References|[4]]]. If the Riemannian metric on  $  M $
has positive scalar curvature, then  $   \mathop{\rm ker}  D = 0 $(
see [[#References|[4]]], [[#References|[5]]]).

A spinor structure on a [[Space-time|space-time]] manifold  $  ( M, g) $(
that is, on a  $  4 $-
dimensional Lorentz manifold) is defined as a spinor structure subordinate to the Lorentz metric  $  g $. 
The existence of a spinor structure on a non-compact space-time  $  M $
is equivalent to the total parallelizability of  $  M $(
see [[#References|[3]]]). As a module over the spinor group  $   \mathop{\rm Spin} ( 1, 3) \approx  \mathop{\rm SL} ( 2, G) $, 
the spinor space decomposes into the direct sum of two complex  $  2 $-
dimensional complexly-conjugate  $   \mathop{\rm SL} ( 2, G) $-
modules  $  {\mathcal C}  ^ {2} $
and  $   {\mathcal C}  dot {}  ^ {2} $. 
This corresponds to the decomposition  $  S( M)= {\mathcal C}  ^ {2} ( M) \oplus  {\mathcal C}  dot {}  ^ {2} ( M) $
of the spinor bundle, where the tensor product  $  {\mathcal C}  ^ {2} ( M) \oplus  {\mathcal C}  dot {}  ^ {2} ( M) $
is identified with the complexification of the tangent bundle  $  TM $. 
Spinor fields in space-time that are eigenfunctions of the Dirac operator characterize free fields of particles with spin  $  1/2 $, 
such as electrons.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Casanova, "L'algèbre vectorielle" , Presses Univ. France  (1976)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  R. Penrose, "The structure of space-time"  C. deWitt (ed.) , ''Batelle Rencontres 1967 Lectures in Math. Physics'' , Benjamin  (1968)  pp. 121–235 (Chapt. VII)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  R. Geroch, "Spinor structure of space-times in general relativity"  ''J. Math. Phys.'' , '''9'''  (1968)  pp. 1739–1744</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  N. Hitchin, "Harmonic spinors"  ''Adv. in Math.'' , '''14'''  (1974)  pp. 1–55</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  A. Lichnerowicz, "Champs spinoriels et propagateurs en relativité générale"  ''Bull. Soc. Math. France'' , '''92'''  (1964)  pp. 11–100</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">  J. Milnor, "Spin structure on manifolds"  ''Enseign. Math.'' , '''9'''  (1963)  pp. 198–203</TD></TR>
<TR><TD valign="top">[7]</TD> <TD valign="top">  R. Penrose, "The twistor programme"  ''Reports Math. Phys.'' , '''12'''  (1977)  pp. 65–76</TD></TR>
<TR><TD valign="top">[8]</TD> <TD valign="top">  R.O., jr. Wells,   "Complex manifolds and mathematical physics"  ''Bull. Amer. Math. Soc.'' , '''1'''  (1979)  pp. 296–336</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Baum,   "Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten" , Teubner  (1981)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  C.T.J. Dodson,   "Categories, bundles, and spacetime topology" , Kluwer  (1988)  pp. Chapt. V, §3 {{ZBL|0661.53016}}</TD></TR>
</table>